The question what would be the best ground for you to request asylum in the US is tricky. The US immigration law doesn’t distinguish some grounds for obtaining asylum from others. In other words, all asylum applicants are treated equally no matter the reason they requested asylum, be it their political opinion, sexual harassment, race, religion, or nationality.
The recent protests and movements against anti-LGBT people may influence the general public opinion. But the growing public sentiment towards people of differing sexual attitudes typically do not influence the decisions taken by the US government with regard to asylum seekers. As a matter of fact, there are many cases when some Americans decide to sponsor people who are representatives of LGBT and requested asylum for the grounds of being persecuted for their sexual orientation. But these Americans can only sponsor asylum seekers who are in custody.
If you’re a refugee, the US government might consider you a danger to the US society and put you behind bars until your asylum case is solved no matter your reasons to seek asylum. The key point is that your case must be supported by some convincing evidence.